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Hyperfine structure in paramagnetic resonance
P h y s i c a X V I I , no 3-4
M a a r t - A p r i l 1951
H Y P E R F I N E S T R U C T U R E IN P A R A M A G N E T I C RESONANCE by B. BLEANEY Clarendon Laboratory, Oxford, England Synopsis
T r a n s i t i o n s are observed at c e n t i m e t r e w a v e l e n g t h s b e t w e e n the Z e e m a n c o m p o n e n t s of the g r o u n d electronic s t a t e of a p a r a m a g n e t i c ion, using a n e x t e r n a l m a g n e t i c field of a few kilogauss. W h e n the nucleus of the ion has a m a g n e t i c m o m e n t , the l a t t e r produces a n a d d i t i o n a l field of a few h u n d r e d gauss acting on the electron. The m a g n i t u d e of this field d e p e n d s on t h e o r i e n t a t i o n of the nucleus, which n o r m a l l y r e m a i n s fixed d u r i n g a n electronic t r a n s i t i o n . The l a t t e r is therefore split into 2 I + 1 c o m p o n e n t s , giving a simple verification of the n u c l e a r spin I. I n the iron group, the d i v a l e n t ions with K r a m e r s degeneracy have been i n v e s t i g a t e d using t u t t o n salts of the type iVIK2(SO4)2, 6H20 , a n d fluosilicates MSiF6, 6H20. To o b t a i n narrow lines, t h e y have been diluted with the corresponding zinc salt in the ratio r o u g h l y of M : Zn ~ 1 : 1000. The results are as follows: Copper, 3d 9, 2D5/2. Isotopes, 63,65. I = 3/2. The orbital m o m e n t u m is largely quenched, leaving a d o u b l e t g r o u n d state, whose spectroscopic splitting factor (g) is anisotropic (in the t u t t o n salts), v a r y i n g from 2.05 to 2.45. The hyperfine s t r u c t u r e is also anisotropic with n e a r l y axial s y m m e t r y ; parallel to the axis the hfs is 4 to 10 times greater t h a n in the p e r p e n d i c u l a r direction, where a more complicated s p e c t r u m is observed due to i n t e r a c t i o n with the electric q u a d r u p o l e of the nucleus. Cobalt, 3d 7, 41~. Isotope, 59. I = 7/2. The g r o u n d s t a t e is a doublet, with v e r y i n c o m p l e t e q u e n c h i n g of the orbital m o m e n t . The g-values are u s u a l l y a b o u t 6 (parallel) a n d 3 (perpendicular). The h y p e r f i n e s t r u c t u r e is also very anisotropic. Manganese, 3d ~, 6S. Isotope, 55. I = 5/2. The splitting of the electronic state (into three doublets) is shown to be due p r i m a r i l y to the n o n - c u b i c s y m m e t r y of the crystalline field, g is isotropic at 2.000 ~ 0.001. The hyperfine s t r u c t u r e is isotropic a n d m u c h larger t h a n expected. V a n a d i u m , 3d 3, 4F. Isotope 51, I = 7/2. The o r b i t a l m o m e n t is q u e n c h e d b y the cubic crystalline field, giving a n --
175
--
176
B. BLEANEY
isotropic g of 1.96. The four-fold spin degeneracy is lifted, two doublets being.left with a splitting of about 0.3 cm - t . The hyperfine structure is again isotropic, and much larger than expected. 1. I n t r o d u c t i o n . The p h e n o m e n o n of p a r a m a g n e t i c resonance m a y be o u t h n e d as follows. The electrons of a p a r a m a g n e t i c ion possess a resultant angular m o m e n t u m , and the ground state consists of a n u m b e r of levels corresponding to the different orientations of the angular m o m e n t u m v e c t o r in an external field. If a magnetic field of a few kilogauss is applied, the separations of these levels correspond to q u a n t a of radiation of centimetre wavelengths. If such radiation is incident on the ion, transitions will be p~oduced when the resonance c o n d i t i o n h v = W ~ - - W m is fulfilled, which can be d e t e c t e d b y the absorption of energy from the radiation. These transitions are induced b y the oscillating magnetic field of the radiation, which m u s t normally be directed at right angles to the external magnetic field. In spectroscopic terms, we are dealing with " m a g n e t i c dipole" transitions. F o r a free ion or atom, the state with magnetic q u a n t u m n u m b e r M has an energy M g f l H in an external magnetic field, and the allowed transitions for our case are M - + M 4- 1. The resonance condition becomes t h e n hv = g f l H
where g is the L a n d ~ splitting factor, a n d just one line is observed. S u b s t i t u t i o n of e h / 4 r l me for the B o h r m a g n e t o n fl shows t h a t the f r e q u e n c y of this line is the same as the L a r m o r precession frequency. In the solid state, the electrons of an a t o m are far from free because t h e y possess electrical charge which is s u b j e c t e d to enormous electric fields from the surrounding atoms. In the first transition group of elements, this interaction is so great t h a t the orbital m o m e n t u m of the electrons is almost completely quenched; t h a t is to say, the orbits are locked into the crystalline electric field and are uninfluenced b y a n y magnetic field which we can produce. There remains the magnetic m o m e n t associated w i t h t h e spin of the electron which is u n a f f e c t e d except through the influence of the spin orbit coupling. This causes a little of the orbital magnetic m o m e n t to be a d m i x e d with the spin so t h a t its b e h a v i o u r in a magnetic field is modified. The principal effects are:
H Y P E R F I N E STRUCTURE IN PARAMAGNETIC RESONANCE
177
a) The'energy associatedwith an external magnetic field m a y still be written as MgflH'for the level M, but g has no longer the ordinary value 2.0023 of a free spin. It m a y b e greater or lower than this, and its value depends generally on the orientation of the magnetic field with respect to the s y m m e t r y axes of the crystalline electric field. K i t t e 11) (1950) has suggested the name "spectroscopic splitting factor" for g in this usage, where it is not the same as the gyromagnetic ratio. b) for values of S greater than ½, the 2S + I levels are generally split up even in zero magnetic field. Magnetic resonance has made a significant contribution to the study of these effects, because it offers a direct method of investigating the behaviour of the energy levels. Previously such information could be obt~'ined only from the measurement of susceptibility and anomalous specific h'eats. These are difficult to interpret, especially in the case of a substance where the unit cell of the crystal lattice contains more than one ion. Usually these ions differ in that their principal magnetic axes, which are dictated b y the local crystal structure, are differently oriented. Susceptibility measurements give only the -net effect of the several ions, whereas in magnetic resonance a distinct spectrum is obtained from each ion, and its individual behaviour investigated. Since the evaluation of the orbital and crystalline field effects requires extended theoretical study, it is convenient to represent the results of experiments in a way which avoids these difficulties. This m a y be done b y defining an "effective spin" S, b y equating the multiplicity of the levels under observation to 2S + 1. It is then found in most cases the observed spectra can be fitted to a Hamiltonian of the form
[-J'= g//fi H, S, + g±fl (H,Sx + H,S,) + D{S2,--~S(S + 1)} (1) Here axial symmetry is assumed about the z-axis; g//and g± are the values of the spectroscopic splitting factor along and perpendicular to this axis. The term. in D represents the effects of an initial splitting due to the crystalline field. In some cases it is necessary to allow for departure from axial symmetry b y the inclusion of terms such as E(S~ - - S~), but often these are small or zero and they will be omitted here for the sake of simplicity, without invalidating any of the general principles. Physica X V I I
12
|78
B. BLEANEY
In strong fields where gfl H >~ D, t h e energy levels m a y be evaluated rather simply. If the external magnetic field makes an angle 0 with the axis of the crystalline field, one finds for the energy of the level with electronic magnetic quantum number M
WM= gflMH + ½D{(M2--½S(S + I)} {~cos2 0--1}
(2)
where the value of g is given b y
g2 .= g#//COS2 0 +-g#l sin20
(3)
For the transition M <--> M - - l, one finds
If, as is usual, the transitions are observed at constant frequency and variable field, this m a y be rewritten as
Thus one obtains a set of 2S lines, equally spaced about a field H 0 = (hv]gfl) ---- 21.4 (~/g) kilogauss, if ; is expressed in cm -1. The spacing of this "fine structure" is greatest along the axis, where it is (2D/gfl) (in field) and it varies as {3~l/g2 cos2 0 - - 1} as the direction of H is changed.
2. Preliminary discussion o/the hyper/ine structure. At
the end of 1948, the:suggestion was put forward b y Professor C. J. G o r t e r 9) that the nuclear magnetic moment might play a role in the splitting of the ground state of a paramagnetic ion through interaction with the magnetic field of the electron shell This field a m o u n t s t o several hundred thousand gauss at the nucleus, and a splitting amounting to 0.01 or 0.1 cm -1 can easily result. Almost immediately afterwards, the presence of such a hyperfine structure was detected b y magnetic resonance in a salt of copper b y the late R. P. P e n r o s e ~), then working at Leiden. At the fields of several kilogauss commonly used for the investigation of paramagnetic resonance, the situation as regards hyperfine structure is similar to that known as the B a c k-G o u d s m i t region in optical spectra, where the magnetic field is strong enough to break down the simple coupling of J and I to a resultant angular
H Y P E R F I N E STRUCTURE I N PARAMAGNETIC RESONANCE
17-9
momentum F *). Instead, the electronic moment takes up one of its 2 J + 1 orientations in the external magnetic field, while the nucleus takes one of its 21 + 1 orientations in the field produced b y the surrounding electrons. As this field amounts to several hundred kilogauss, the direct effect of the external field on the nucleus m a y be neglected in comparison. When the application of an r.f. magnetic field causes a transition, the electronic moment changes its orientation, but the nuclear moment does not, since the effect of the r.f. field on the tiny nuclear moment is negligible. The change of orientation of the electronic moment causes a change in the field which acts on the nucleus, a n d hence a change in the energy of interaction. This energy change differs for each of the 21 + l orientations, and since the energy difference must be supplied b y the radiation quantum, it follows thai the spectrum line will be split into 21 + 1 components. These components are equally spaced, since the electronicnuclear interaction energy is proportional to the product Mm, where M and m are the magnetic quantum numbers of electron and nucleus respectively. The hfs splitting is the same for all transitions for which A M - ~ 4- I, A m = O . The situation which arises when the his is investigated at constant frequency and variable magnetic field can be understood from fig. 1. This illustrates the ease of copper, for which we have I ~- 3/2, J = S (effectively) = ½. The two levels M = +½ a n d i are each sprit into four components of equal separation; these components are inverted in one case, and upright in the other, since the field at the nucleus reverses as the electron turns over. As the external field is increased, the energy of the allowed transitions passes in turn through the resonant value by, and the spectrum observed is again one of four Components, equally spaced in magnetic field. The shift from the centre of the pattern is just equal to the mean magnetic field which the nucleus in its given orientation e x e c s on the electron; this field is added to the external field thus ca us'mg the transition to occur at a different external field for each nuclear orientation. It should be noted that this simple picture applies only when the separation between successive electronic levels is large compared with the hfs splitting; that is, when the external field is large compared with the field which the nucleus exerts on the electron. *) I n the solid state F is not.a good quantum number, even in the sense F = S + I , because of the crystalline field affects (anisotropy etc.).
180
B. BLEANEY
The Hamiltonian for the magnetic interaction between electron and nucleus m a y be written in the usual form z~/'=27fl~vfl 7
I.L--s.I-~
r2
.
Im
S-I
J J J
! h~ I
÷~
+÷
÷~
1
2
3
2
H Fig. 1. Consideration of this formula shows that one would expect the hfs to be anisotropic; that is to say, the separations between lines of the hfs will vary with the direction of the external magnetic field relative to the crystalline axes. This anisotropy arises in two ways; (1) the
H Y P E R F I N E STRUCTURE IN PARAMAGNETIC RESONANCE
181
amount of residual orbital moment admixed by the spin-orbit coupling depends on the direction of the external field; (2) the coupling between the electron spin and the nuclear spin depends on their orientation in the characteristic manner of dipole-dipole coupling. This coupling (2) must be averaged over the electron distribution in the occupied orbit, the latter being determined b y the crystalline electric field. Thus both the orbital and spin contributions to the hfs are affected by the crystalline field, and it follows that the anisotropy will have the same axes of symmetry as the crystalline field and the electronic spectroscopic splitting factor g. In the interpretation of the experimental data, the separation of the contributions of orbit and spin requires long and difficult theoretical study. It is therefore natural to analyse the spectra in terms of a simple Hamiltonian which m a y be written
AS:I, + B(Sj, + S,I,). Here the anisotropy is assumed to have axial symmetry (about the z-axis), as this is found to hold closely in all the spectra investigated. As previously, the components of the electronic angular momentum are written as those of a spin, S. The values of A, B become then parameters to be determined experimentally from the observed spectra and their analysis. In addition to the magnetic interaction between the electrons and the nucleus, there m a y be a coupling through the nuclear electric quadrupole moment and the gradient of the electric field of the electrons at the nucleus. This gradient is zero if the electron cloud distribution has spherical or cubic symmetry, but in one case (copper) the electric coupling is comparable with the magnetic coupling. The effects m a y be represented by an additional term
Q {I~ --~I(I + 1)} in the Hamiltonian. Here Q is again a parameter to be determined experimentally. If one includes also the direct effect of the field H on the nucleus, one obtains altogether
IT-~= AS,I,+B(S=I=+SyI,) + Q {I~ - - } I ( I + l)} - - 7flNH. I
(5)
which is to be added to the expression given by (1). In a strong external field at an angle 0 to the z-axis, the energy of the level M , m is now given by adding to (2) the expression
182
B. BLEANEY
{' (A2~//cos20 + B2g2 sin20)½} Mm + }O {m2--{I(I+ 1)}.
W~,,.= g
f ~A2~//cos2 e __l}__TflNHm(Ag// •/3 \-~@ cos2e+ ~-~g~sin20)
(6)
where 1. g Then for the transition (M, m ) ~
K = - - ( A 2 ~ cos ~ e + B 2 d i sin 2 e ) ~.
( M - - l , m) one must add to (4)
Km
(7)
and to (4a)
o
/7,,
It will be seen that the effects of the quadrupole term and the direct interaction between nucleus and field on the spectrum are zero, and each electronic transition is split up into 21 + 1 lines of equal separation in field given by the coefficient of m in (7a). The absence of observable quadrupole effects holds only for the case of Q < B and this point merits some further discussion. The quadrupole interaction produces a first order shift in the energy level (6), but this shift is (to this order) the same in each electronic state for a given m. Thus no first order effect on the spectrum *) is observed so long as the selection rule Am = 0 is obeyed. This selection nile breaks down in the case when Q ~ B, and the external magnetic field makes a large angle with the symmetry axis. The reason for this is that the quadrupole effect tries to align the nucleus along the symmetry axis, while the magnetic field of the electrons is trying to align it almost at right angles. When the external magnetic field is exactly perpendicular to the symmetry axis, the total number of possible lines becomes 6 1 - l, whose intensities are not all the same. The spectrum is therefore much more complicated than in the absence of quadrupole effects. 3. Hyper/ine structure in the iron group. Investigations at Oxford have been largely confined to the ions of nuclei, whose isotopes are 100% odd and should therefore all contribute a hyperfine structure, provided that the magnetic field of the electrons at the nucleus is *) T h e s p e c t r a l lines are shifted in the s e c o n d order b y a m o u n t ~
(q2/B).
HYPERFINE STRUCTURE IN PARAMAGNETIC RESONANCE
183
finite. There are four such ions, namely vanadium, manganese, cobalt and copper, which have been studied as divalent ions in either tutton salts, such as Mn(NH4)2(SO,)2, 6H20 or fluosilicates MnSiF6, 6H20. The former have two ions in unit cell, each with approximately tetragonal symmetry; the latter have only one, with tfigonal symmetry. To obtain narrow lines, highly "diluted" salts in the form of mixed crystals with the isomorphous zinc salt, have been grown. At concentrations of the order of one paramagnetic ion to about a thousand zinc ions, a limiting half-width at half intensity of about 7 gauss is attained. This residual width is attributed to the twelve protons in the waters of crystallization which lie immediately around the paramagnetic ion. The hyperfine structure of these four ions contained significant differences in behaviour, and it is convenient to consider them separately. Manganese. 3d s, 6S, I s/2. The manganese ion serves as a test point of theory since the 3d shell is just half-filled, so that the free ion is in a s S state and no crystalline field is required to "quench" the orbital motion. As a result, the value of g (2.000 4- 0.001 in the fluosilicate) is very close to the free spin value (2.0023,) and exceedingly isotropic. A small splitting 4) of the 6S state is found, corresponding to values of D respectively of + 0.027 cm -1 (manganese ammonium sulphate) and --0.013 to --0.018 cm - I (fluosilicate, depending on temperature). These splittings are attributed to dipole-dipole interaction between the spin moments of the five 3d electrons; this interaction is zero if the electron cloud has spherical symmetry, as in the free ion, but is not zero if a small axial distortion is present due to the crystalline field 5). As a consequence of this spherical symmetry, the magnetic field and electric field gradient at the nucleus would be zero for the free ion, and no his would result. The small distortion in the solid could produce only a tiny his, with A = - - 2B, and both of the order of 10-4 cm -L. In fact, an almost isotropic his is observed, of magnitude one hundred times greater. It is practically identical in magnitude in the two salts, showing that it is not due to the crystalline field, since the electronic splitting shows this differs in magnitude (and sign) in the two salts. A b r a g a m e ) has pointed out that an his of this magnitude must be due to an unpaired s-electron; he suggests that =
184
B. B L E A N E Y
the .3s23p63d~, 6S state h a s admixed it some of a 3s3d63dS4s, 6S state, in which a 3s electron is promoted to the 4s level. Only a very small admixture of such a state is required, since a 3s electron is very plotent in producing hyperfine structure. The admixture is due to the electrostatic repulsion of the electrons (configurational interaction), Which is ~, property of the free ion.,This explains why the effect is the same in two very different salts.
Vanadium, 3d a, I =
7/2.
H e r e the free ion is in a 4/~ state, but a crystalline field of cubic symmetry quenches the orbit very effectively. The lowest orbital level is a singlet, some 10,000 cm ~l below any other orbital level. The value of g is 1.96, and is isotropic. This lowest level is fourfold in the Spin, but is split into two doublets (in zero magnetic field) b y second order spin-orbit interaction and the spin-spin interaction described far manganese. In vanadium ammonium sulphate this splitting corresponds to D----0.16 cm -I, but there is also a very considerable rhombic contribution. The wave function of the lowest orbit is xyz, which has six equal lobes along the rectangular axes, This leads to zero magnetic field at the nucleus from the electron spins, as m a y easily be seen classically b y computing the field at the origin due to six dipoles placed at equal distanoes along the axes. There remains only the orbital contribution, which s h o u l d b e small as the departure of g from the free spin values is small. Experimentally a wide and iso~ropic his is observed, with A = B = 0.01 cm -I, which is practically the same as in manganese. Though this is only a coincidence, there is no doubt that the main contribution to this his comes again from admixture with a state in wich a 3s electron is promoted. The spin of 7/2 (K o pf e r m a n n and R a s m u s s e n S ) ) , which is given with a query in some tables, is confirmed. Some preliminary work has been carried out to determine the spin of chromium 53, which is present in 9 o abundance in the natural state, using a diluted chrome alum. Here the chromium trivalent ion isiso-electronic with divalent vanadium. No h/s could be observed and it seems probable that the overall width of any h/s must be less than 60 gauss, as compared with 680 in vanadium. This would suggest that the nuclear moment of Cr~ is less than 0.5 nuclear magnetons, but further work is in hand to confirm this result,
HYPERFINE
STRUCTURE
IN PARAMAGNETIC
RESONANCE
185
Cobalt, 3d 7. Co59~I, = 7/2. As in the case of V ++ , the state of the tree ion is 4F, but as we are dealing with three holes in a full shell instead of three electrons, the splitting of the orbital levels of cobalt b y the major (Cubic) portion of the crystalline field is inverted. The cubic field leaves an orbital triplet at t h e bottom, which is split .up b y spin-orbit interaction and tetragdla~(~(:or trigonal) crystalline field 9) to a' rather small degree. The lowest is a K r a m e r's doublet, a n d t h e next level lies only a few hundred cm -1 higher. This accounts for the great anisotropy shown b y cobalt salts, whose susceptibility also departs widely from C u r i e ' s law. The results obtained with three salts are summarized b y the values lo) given in the following table. ,, TABLE I g-values of Co
.
A , cm - t
B em -~
3.44
0.018
0.005
6.55
2.4-3.3
0.029.
0.007
6.45
3.06
0.025
0.002
Salt
g#
C o b a l t fluosilicate . . Cobalt potassium sulphate Cobalt ammonium sulphate .....
5.82
g±
The variation in g± shown for the potassium sulphateis due to the presence of rhombic asymmetry; the other salts have very accurately axial symmetry. It will be seen that both the g-values and the h/s show considerable anisotropy. P r y c e and A b r a g a m n), in an extensive study of cobalt, have shown how the theory which accounts for the g-values gives also a reasonable approximation to the hyperfine structure. The latter is due almost entirely to interaction with the orbital moment, the spin contribution being only about 10°/o. Some interaction with a state in which a 3s electron is promoted is required to explain the high anisotropy, but the effect of such a state is relatively small. Copper, 3d 9, 2 D . C u 63, C u 6s, I ~--- 3/2. In a field of tetragonal symmetry, as in the tutton salts, the orbital m o m e n t u m of Cu + + is largely quenched, as is shown b y the values of g. ParaUel to the axis, g is 2.4---2.5; perpendicular, g is 2.05--2.1. This suggests that the his will be due primarily to interaction with
186
B. BLEANEY
the electron s p i n , and at first sight the measured anisotropy (A ---- 0.012 cm - l , B ---- 0.001 to 0.003 cm -1, varying with the salt) supports this. The lowest orbital wavefunction is ( ~ - y2), which has four equal lobes along the two axes normal to the tetragonal axis. If four dipoles are placed at equal distances along these axes, it is easily seen that the field which they produce at the origin is twice as great when the dipoles are aligned parallel to the z-axis, as it is when they are aligned in the xy-plane. As there is also a change of sign, one would expect A ~ - - 2 B ; this appears to fit the results reasonably, since the signs are indeterminate (see § 5). A b r a g a m and P r y c e 12,e) have shown however that the contribution from the unquenched orbit upsets this pseudo-agreement completely, giving A = - - 0 . 0 0 6 cm - I , B = 0.013 cm -I. Theory can only be reconciled with experiment b y admixing excited configurations; 3d84s 2D gives a contribution whose sign is ~uch as to worsen the situation, but 3s 3p63d94s, 2/9 provides the right correction. A b r agame) shows that one can obtain then A - ~ - - 0 . 0 1 5 cm -1, B ~ + 0.004 cm -1 in reasonable agreement with experiment. Thus again a significant feature of the theory is that a state in which an inner 3s electron is promoted to a 4s must be considered. Although copper has two abundant isotopes, their magnetic moments differ only b y 7%, and their his shows only partial resolution when it is widest, near the axis. I n g r a m 18) discovered, however, that the spectrum is split into more than four lines, when the magnetic field is nearly perpendicular to the axis. P r y c e has shown that the general behaviour can be fitted b y assuming the existence of a quadrupole effect with Q ~ 0.001 c m - ' . It is hoped b y further work on this spectrum to obtain more exact information of the size of Q for the two isotopes, 63 and 65. As this is the only case in which a quadrupole effect has been detected, it should be pointed out that this does not imply that the electric quadrupole moments of vanadium, manganese and cobalt are necessarily much smaller than those of the copper isotopes. It is obvious that the electron cloud distribution for V and Mn is too symmetrical to give an appreciable gradient of the electric field at the nucleus, and the admixture of an excited state with an unpaired s-electron does. nothing to change this situation. In cobalt also, theory indicates that any quadrupole interaction would be small.
HYPERFINE
STRUCTURE
IN
PARAMAGNETIC
RESONANCE
187
From the experiments only the following upper limits can be set for Q : vanadium, manganese (divalent salts), 0.00 1 cm-’ cobalt (divalent salts), 0.0005 cm-‘. 4. Hyfierfine structure in the rare earth group. Little paramagnetic resonance work has been carried out so far on salts of the rare earth group, but a start has recently been made at Oxford. One of the most suitable salts seems to be the ethyl sulphates, which have trigonal symmetry, and, from the magnetic point of view, only one ion in unit cell. These are already, magnetically, fairly dilute, and can be diluted further by the isomorphous salt of lanthanum. The results obtained thus with nedodymium and gadolinium are given below. The rare earth group differs from the iron groups in that L and S still combine to form a resultant J, and the crystalline field is strong enough only to split up the 2J + 1 levels corresponding to the different orientations of J. These splittings are thought to be of the order of IO-1 00 cm-‘. Although,when J is half integral, the crystalline field must leave at least double degeneracy in the levels, it does not necessarily follow that resonance absorption can be obtained in the region V = 1 cm-‘. The degenerate levels may correspond, for example to J, = + 3/2 and - 3/2,between which no transitions are allowed. However, it seems probable, that by choice of a salt with suitable crystalline field, absorption lines may be obtained in all these cases, This is important, for the trivalent ions with K r am e r s degeneracy are cerium, neodymium, samarium, gadolinium dysprosium, erbium and ytterbium. Though these elements have even atomic number, all except cerium have one or two odd isotopes of natural abundance 1O-2Oo/o,and the spins of seven of these are not previously known. Work has therefore been concentrated on these, starting with neodymium and gadolinium l*). Neodymizrm 4f3. Odd isotopes : 143, 12.2% abundance. 145, 8.3% abundance. Even isotopes : 79 y0 abundance. An extensive hyperfine structure in concentrated neodymium ethyl sulphate was reported by B 1 e a n e y and I n g r a m lo). At a dilution of Nd : La w 1 : 200, the half width of the lines is reduced to about 7 gauss. The hyperfine structure is then completely resolved, and two series each containing eight equally spaced lines
188
B. B L E A N E Y
can be assigned to the odd isotopes 143, 145. F r o m t h e spacing the ratio of the m a g n e t i c m o m e n t s (143/145) is f o u n d to be 1.61 + 0.01. T h e d a t a m a y be s u m m e d up as follows. g// =
3.58,
gL
: 2.09
Isotope 143 : A = 0.040 c m -1, B = 0.020 cm - t , I = 7/2. Isotope 145: A = 0.025 cm -1, B = 0.012 cm - j , I = 7/2. T h u s the anisotropy is 2 : 1 in the hyperfine structure, a n d a b o u t 1.7 : I in the g-values. No effects due to a q u a d r u p o l e interaction could be observed, a n d an upper limit for {) of 0.002 cm -I can be assigned for either isotope. Gadolinium, 4/7. Odd isotopes: 155, 15% aburndance. 15•, 15% abundance. E v e n isotopes: 70% abundance. The trivalent gadolinium ion is in an 8S state, and a fine s t r u c t u r e is observed due to a small splitting of the levels. Like manganese, this splitting is due primarily to an axial distortion ; the value of t h e p a r a m e t e r D is a p p r o x i m a t e l y 0.023 cm -1, varying by about 105/o between room t e m p e r a t u r e and 20°K. Although t h e signal from each line of the fine s t r u c t u r e was 50 to 100 times noise, no weak lines were visible belonging to h y p e r f i n e structure. As a f u r t h e r check the s p e c t r u m was observed at an angle where 5 of t h e fine s t r u c t u r e lines coincide; since a n y hfs belonging to these transitions would also coincide. This should give a gain of five in intensity; its absence showed conclusively t h a t there was no resolved h/s. The overall w i d t h of a n y his m u s t be less t h a n 40 gauss, c o m p a r e d with a maxim u m w i d t h of 1600 gauss in n e o d y m i u m . This absence of hfs shows t h a t t h e g r o u n d state of the gadolinium ion is exceedingly free from a d m i x t u r e of a n y excited states in which, for instance, a 4s or 5s electron has been p r o m o t e d to the 6s level. T h u s the situation is v e r y different from the iron group, where the interaction w i t h such excited states is sufficient to produce an his in v a n a d i u m a n d manganese (where there should be none) of the same order or greater t h a n in cobalt and copper. As a result, no information concerning the nuclei of the odd gadolinium isotopes could be obtained.
5. Conclusion. At t h e end of this brief review of the work on his in the solid state carried out at the Clarendon L a b o r a t o r y in the past 20 months, it seems appropriate to discuss the possibilities of ob-
H Y P E R F I N E STRUCTURE IN P A R A M A G N E T I C RESONANCE
189
taining nuclear information b y this method. It is, of course, limited to atoms which form paramagnetic salts, but it is just these which are rather inaccessible to other methods. It is obvious that a simple and direct determination of nuclear spin is possible, together with a fairly accurate measure of the ratio of magnetic moments of two isotopes of the same atom. The h/s splitting can be measured with an accuracy (at present) of about 1% ; this could be improved b y the use of a proton resonance calibration for the magnetic field. There seems little gain in such improvement, however, because of the difficulties of the theoretical interpretation. It is well known that high accuracy is not obtainable in the calculation of absolute values of nuclear moments from optical h/s; here one has the additional complication of the crystalline .field. In the iron group, it appears' :that configurational" interaction plays a very important role, and the theory which is adequate to explain the g-values tells us practically nothing about this effect. In the rare earth group, the absence of h/s in gadolinium suggests that this effect may be much less troublesome, b u t here the theory of the g-values is rudimentary. K. W. H. S t e v e n s 16) has tackled the neodymium ion, and we may hope that when a suitablecombination of crystalline field and spin-orbit interactions has been found to fit the g-values, a reliable estimate of the nuclear moments m a y also be obtained. A regrettable feature of the method is that it gives no information about the actual signs of the constants A, B, Q. Where several electronic transitions are observed, as in manganese, the relative signs A, B and D can be found from second order effects of the spectrum in not quite strong field; B 1 e a n e y and I n g r a m 4) have combined this with a measurement of the anisotropy of the susceptibility which gives the sign of D, to obtain all the signs. But when S = ½, not even the relative signs of A, B can be found in the simple case. Where a quadrupole effect is observed, as in copper, the extra transitions can m theory be used to find the relative signs, when allowance is made for the direct effect of the external field on the nucleus. The second order effects in copper potassium sulphate suggests that B, q have the same sign, b u t even here a previous knowledge of the sign of nuclear moment is necessary. More accurate information about the size of q, and its sign relative to A can be obtained from observations of the spectrum in .zero or very small magnetic field, and work in the region of 50--100 cm wavelength is planned. Received 23-9-50.
] 90
HYPERFINE STRUCTURE IN PARAMAGNETIC RESONANCE REFERENCES
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) IS)
K i t t e l , C.H., Phys. Rev. 76 (1949) 743. Gorter, C . J . , Physica, 14 (1948) 504. Penrose, R . P . , Nature Lond., 163 (1949) 992. Bleaney, B. and I u g r a m , D. J. E. Proc. roy. Soc. A205(1951) 336. Abraham, A. and P r y e e , M . H . L . , P r o c . roy. Soc. A'205(1951) 135. Abragam, A., Phys. Rev. 79 (1950) 534. I n g r a i n , D . J . E . a~d S e o v i l , H . E . D . , (to be published). Kopfermann, H. and R a s m u s s e n , E., Z. Physik, 98 (1936)624. P r y c e , M . H . L . , NatureLond, 164(1949) 116. Bleaney, B..and I n g r a m , D . J . E . , NatureLond. 164(1949) l l6 and future publication. Abragam, A. and P r y c e , M. H. L. (to be published). Abragam, A. and P r y c e , M . H . L . , Nature Loud. 163 (1949) 992. I n g r a i n , D. J. E., Prox. Phys. Soe. A6'2 ¢q949) 664. Bleaney, B. and S e o v i l , H . E . D . , Proe. phys. Soc. A 63 (1950)1369. Elliott, R. J. and S t e v e n s , K . W . H . , P r o e . phys. Soc. A64(1951) 205.
M a a r t - A p r i l 1951
H Y P E R F I N E S T R U C T U R E IN P A R A M A G N E T I C RESONANCE by B. BLEANEY Clarendon Laboratory, Oxford, England Synopsis
T r a n s i t i o n s are observed at c e n t i m e t r e w a v e l e n g t h s b e t w e e n the Z e e m a n c o m p o n e n t s of the g r o u n d electronic s t a t e of a p a r a m a g n e t i c ion, using a n e x t e r n a l m a g n e t i c field of a few kilogauss. W h e n the nucleus of the ion has a m a g n e t i c m o m e n t , the l a t t e r produces a n a d d i t i o n a l field of a few h u n d r e d gauss acting on the electron. The m a g n i t u d e of this field d e p e n d s on t h e o r i e n t a t i o n of the nucleus, which n o r m a l l y r e m a i n s fixed d u r i n g a n electronic t r a n s i t i o n . The l a t t e r is therefore split into 2 I + 1 c o m p o n e n t s , giving a simple verification of the n u c l e a r spin I. I n the iron group, the d i v a l e n t ions with K r a m e r s degeneracy have been i n v e s t i g a t e d using t u t t o n salts of the type iVIK2(SO4)2, 6H20 , a n d fluosilicates MSiF6, 6H20. To o b t a i n narrow lines, t h e y have been diluted with the corresponding zinc salt in the ratio r o u g h l y of M : Zn ~ 1 : 1000. The results are as follows: Copper, 3d 9, 2D5/2. Isotopes, 63,65. I = 3/2. The orbital m o m e n t u m is largely quenched, leaving a d o u b l e t g r o u n d state, whose spectroscopic splitting factor (g) is anisotropic (in the t u t t o n salts), v a r y i n g from 2.05 to 2.45. The hyperfine s t r u c t u r e is also anisotropic with n e a r l y axial s y m m e t r y ; parallel to the axis the hfs is 4 to 10 times greater t h a n in the p e r p e n d i c u l a r direction, where a more complicated s p e c t r u m is observed due to i n t e r a c t i o n with the electric q u a d r u p o l e of the nucleus. Cobalt, 3d 7, 41~. Isotope, 59. I = 7/2. The g r o u n d s t a t e is a doublet, with v e r y i n c o m p l e t e q u e n c h i n g of the orbital m o m e n t . The g-values are u s u a l l y a b o u t 6 (parallel) a n d 3 (perpendicular). The h y p e r f i n e s t r u c t u r e is also very anisotropic. Manganese, 3d ~, 6S. Isotope, 55. I = 5/2. The splitting of the electronic state (into three doublets) is shown to be due p r i m a r i l y to the n o n - c u b i c s y m m e t r y of the crystalline field, g is isotropic at 2.000 ~ 0.001. The hyperfine s t r u c t u r e is isotropic a n d m u c h larger t h a n expected. V a n a d i u m , 3d 3, 4F. Isotope 51, I = 7/2. The o r b i t a l m o m e n t is q u e n c h e d b y the cubic crystalline field, giving a n --
175
--
176
B. BLEANEY
isotropic g of 1.96. The four-fold spin degeneracy is lifted, two doublets being.left with a splitting of about 0.3 cm - t . The hyperfine structure is again isotropic, and much larger than expected. 1. I n t r o d u c t i o n . The p h e n o m e n o n of p a r a m a g n e t i c resonance m a y be o u t h n e d as follows. The electrons of a p a r a m a g n e t i c ion possess a resultant angular m o m e n t u m , and the ground state consists of a n u m b e r of levels corresponding to the different orientations of the angular m o m e n t u m v e c t o r in an external field. If a magnetic field of a few kilogauss is applied, the separations of these levels correspond to q u a n t a of radiation of centimetre wavelengths. If such radiation is incident on the ion, transitions will be p~oduced when the resonance c o n d i t i o n h v = W ~ - - W m is fulfilled, which can be d e t e c t e d b y the absorption of energy from the radiation. These transitions are induced b y the oscillating magnetic field of the radiation, which m u s t normally be directed at right angles to the external magnetic field. In spectroscopic terms, we are dealing with " m a g n e t i c dipole" transitions. F o r a free ion or atom, the state with magnetic q u a n t u m n u m b e r M has an energy M g f l H in an external magnetic field, and the allowed transitions for our case are M - + M 4- 1. The resonance condition becomes t h e n hv = g f l H
where g is the L a n d ~ splitting factor, a n d just one line is observed. S u b s t i t u t i o n of e h / 4 r l me for the B o h r m a g n e t o n fl shows t h a t the f r e q u e n c y of this line is the same as the L a r m o r precession frequency. In the solid state, the electrons of an a t o m are far from free because t h e y possess electrical charge which is s u b j e c t e d to enormous electric fields from the surrounding atoms. In the first transition group of elements, this interaction is so great t h a t the orbital m o m e n t u m of the electrons is almost completely quenched; t h a t is to say, the orbits are locked into the crystalline electric field and are uninfluenced b y a n y magnetic field which we can produce. There remains the magnetic m o m e n t associated w i t h t h e spin of the electron which is u n a f f e c t e d except through the influence of the spin orbit coupling. This causes a little of the orbital magnetic m o m e n t to be a d m i x e d with the spin so t h a t its b e h a v i o u r in a magnetic field is modified. The principal effects are:
H Y P E R F I N E STRUCTURE IN PARAMAGNETIC RESONANCE
177
a) The'energy associatedwith an external magnetic field m a y still be written as MgflH'for the level M, but g has no longer the ordinary value 2.0023 of a free spin. It m a y b e greater or lower than this, and its value depends generally on the orientation of the magnetic field with respect to the s y m m e t r y axes of the crystalline electric field. K i t t e 11) (1950) has suggested the name "spectroscopic splitting factor" for g in this usage, where it is not the same as the gyromagnetic ratio. b) for values of S greater than ½, the 2S + I levels are generally split up even in zero magnetic field. Magnetic resonance has made a significant contribution to the study of these effects, because it offers a direct method of investigating the behaviour of the energy levels. Previously such information could be obt~'ined only from the measurement of susceptibility and anomalous specific h'eats. These are difficult to interpret, especially in the case of a substance where the unit cell of the crystal lattice contains more than one ion. Usually these ions differ in that their principal magnetic axes, which are dictated b y the local crystal structure, are differently oriented. Susceptibility measurements give only the -net effect of the several ions, whereas in magnetic resonance a distinct spectrum is obtained from each ion, and its individual behaviour investigated. Since the evaluation of the orbital and crystalline field effects requires extended theoretical study, it is convenient to represent the results of experiments in a way which avoids these difficulties. This m a y be done b y defining an "effective spin" S, b y equating the multiplicity of the levels under observation to 2S + 1. It is then found in most cases the observed spectra can be fitted to a Hamiltonian of the form
[-J'= g//fi H, S, + g±fl (H,Sx + H,S,) + D{S2,--~S(S + 1)} (1) Here axial symmetry is assumed about the z-axis; g//and g± are the values of the spectroscopic splitting factor along and perpendicular to this axis. The term. in D represents the effects of an initial splitting due to the crystalline field. In some cases it is necessary to allow for departure from axial symmetry b y the inclusion of terms such as E(S~ - - S~), but often these are small or zero and they will be omitted here for the sake of simplicity, without invalidating any of the general principles. Physica X V I I
12
|78
B. BLEANEY
In strong fields where gfl H >~ D, t h e energy levels m a y be evaluated rather simply. If the external magnetic field makes an angle 0 with the axis of the crystalline field, one finds for the energy of the level with electronic magnetic quantum number M
WM= gflMH + ½D{(M2--½S(S + I)} {~cos2 0--1}
(2)
where the value of g is given b y
g2 .= g#//COS2 0 +-g#l sin20
(3)
For the transition M <--> M - - l, one finds
If, as is usual, the transitions are observed at constant frequency and variable field, this m a y be rewritten as
Thus one obtains a set of 2S lines, equally spaced about a field H 0 = (hv]gfl) ---- 21.4 (~/g) kilogauss, if ; is expressed in cm -1. The spacing of this "fine structure" is greatest along the axis, where it is (2D/gfl) (in field) and it varies as {3~l/g2 cos2 0 - - 1} as the direction of H is changed.
2. Preliminary discussion o/the hyper/ine structure. At
the end of 1948, the:suggestion was put forward b y Professor C. J. G o r t e r 9) that the nuclear magnetic moment might play a role in the splitting of the ground state of a paramagnetic ion through interaction with the magnetic field of the electron shell This field a m o u n t s t o several hundred thousand gauss at the nucleus, and a splitting amounting to 0.01 or 0.1 cm -1 can easily result. Almost immediately afterwards, the presence of such a hyperfine structure was detected b y magnetic resonance in a salt of copper b y the late R. P. P e n r o s e ~), then working at Leiden. At the fields of several kilogauss commonly used for the investigation of paramagnetic resonance, the situation as regards hyperfine structure is similar to that known as the B a c k-G o u d s m i t region in optical spectra, where the magnetic field is strong enough to break down the simple coupling of J and I to a resultant angular
H Y P E R F I N E STRUCTURE I N PARAMAGNETIC RESONANCE
17-9
momentum F *). Instead, the electronic moment takes up one of its 2 J + 1 orientations in the external magnetic field, while the nucleus takes one of its 21 + 1 orientations in the field produced b y the surrounding electrons. As this field amounts to several hundred kilogauss, the direct effect of the external field on the nucleus m a y be neglected in comparison. When the application of an r.f. magnetic field causes a transition, the electronic moment changes its orientation, but the nuclear moment does not, since the effect of the r.f. field on the tiny nuclear moment is negligible. The change of orientation of the electronic moment causes a change in the field which acts on the nucleus, a n d hence a change in the energy of interaction. This energy change differs for each of the 21 + l orientations, and since the energy difference must be supplied b y the radiation quantum, it follows thai the spectrum line will be split into 21 + 1 components. These components are equally spaced, since the electronicnuclear interaction energy is proportional to the product Mm, where M and m are the magnetic quantum numbers of electron and nucleus respectively. The hfs splitting is the same for all transitions for which A M - ~ 4- I, A m = O . The situation which arises when the his is investigated at constant frequency and variable magnetic field can be understood from fig. 1. This illustrates the ease of copper, for which we have I ~- 3/2, J = S (effectively) = ½. The two levels M = +½ a n d i are each sprit into four components of equal separation; these components are inverted in one case, and upright in the other, since the field at the nucleus reverses as the electron turns over. As the external field is increased, the energy of the allowed transitions passes in turn through the resonant value by, and the spectrum observed is again one of four Components, equally spaced in magnetic field. The shift from the centre of the pattern is just equal to the mean magnetic field which the nucleus in its given orientation e x e c s on the electron; this field is added to the external field thus ca us'mg the transition to occur at a different external field for each nuclear orientation. It should be noted that this simple picture applies only when the separation between successive electronic levels is large compared with the hfs splitting; that is, when the external field is large compared with the field which the nucleus exerts on the electron. *) I n the solid state F is not.a good quantum number, even in the sense F = S + I , because of the crystalline field affects (anisotropy etc.).
180
B. BLEANEY
The Hamiltonian for the magnetic interaction between electron and nucleus m a y be written in the usual form z~/'=27fl~vfl 7
I.L--s.I-~
r2
.
Im
S-I
J J J
! h~ I
÷~
+÷
÷~
1
2
3
2
H Fig. 1. Consideration of this formula shows that one would expect the hfs to be anisotropic; that is to say, the separations between lines of the hfs will vary with the direction of the external magnetic field relative to the crystalline axes. This anisotropy arises in two ways; (1) the
H Y P E R F I N E STRUCTURE IN PARAMAGNETIC RESONANCE
181
amount of residual orbital moment admixed by the spin-orbit coupling depends on the direction of the external field; (2) the coupling between the electron spin and the nuclear spin depends on their orientation in the characteristic manner of dipole-dipole coupling. This coupling (2) must be averaged over the electron distribution in the occupied orbit, the latter being determined b y the crystalline electric field. Thus both the orbital and spin contributions to the hfs are affected by the crystalline field, and it follows that the anisotropy will have the same axes of symmetry as the crystalline field and the electronic spectroscopic splitting factor g. In the interpretation of the experimental data, the separation of the contributions of orbit and spin requires long and difficult theoretical study. It is therefore natural to analyse the spectra in terms of a simple Hamiltonian which m a y be written
AS:I, + B(Sj, + S,I,). Here the anisotropy is assumed to have axial symmetry (about the z-axis), as this is found to hold closely in all the spectra investigated. As previously, the components of the electronic angular momentum are written as those of a spin, S. The values of A, B become then parameters to be determined experimentally from the observed spectra and their analysis. In addition to the magnetic interaction between the electrons and the nucleus, there m a y be a coupling through the nuclear electric quadrupole moment and the gradient of the electric field of the electrons at the nucleus. This gradient is zero if the electron cloud distribution has spherical or cubic symmetry, but in one case (copper) the electric coupling is comparable with the magnetic coupling. The effects m a y be represented by an additional term
Q {I~ --~I(I + 1)} in the Hamiltonian. Here Q is again a parameter to be determined experimentally. If one includes also the direct effect of the field H on the nucleus, one obtains altogether
IT-~= AS,I,+B(S=I=+SyI,) + Q {I~ - - } I ( I + l)} - - 7flNH. I
(5)
which is to be added to the expression given by (1). In a strong external field at an angle 0 to the z-axis, the energy of the level M , m is now given by adding to (2) the expression
182
B. BLEANEY
{' (A2~//cos20 + B2g2 sin20)½} Mm + }O {m2--{I(I+ 1)}.
W~,,.= g
f ~A2~//cos2 e __l}__TflNHm(Ag// •/3 \-~@ cos2e+ ~-~g~sin20)
(6)
where 1. g Then for the transition (M, m ) ~
K = - - ( A 2 ~ cos ~ e + B 2 d i sin 2 e ) ~.
( M - - l , m) one must add to (4)
Km
(7)
and to (4a)
o
/7,,
It will be seen that the effects of the quadrupole term and the direct interaction between nucleus and field on the spectrum are zero, and each electronic transition is split up into 21 + 1 lines of equal separation in field given by the coefficient of m in (7a). The absence of observable quadrupole effects holds only for the case of Q < B and this point merits some further discussion. The quadrupole interaction produces a first order shift in the energy level (6), but this shift is (to this order) the same in each electronic state for a given m. Thus no first order effect on the spectrum *) is observed so long as the selection rule Am = 0 is obeyed. This selection nile breaks down in the case when Q ~ B, and the external magnetic field makes a large angle with the symmetry axis. The reason for this is that the quadrupole effect tries to align the nucleus along the symmetry axis, while the magnetic field of the electrons is trying to align it almost at right angles. When the external magnetic field is exactly perpendicular to the symmetry axis, the total number of possible lines becomes 6 1 - l, whose intensities are not all the same. The spectrum is therefore much more complicated than in the absence of quadrupole effects. 3. Hyper/ine structure in the iron group. Investigations at Oxford have been largely confined to the ions of nuclei, whose isotopes are 100% odd and should therefore all contribute a hyperfine structure, provided that the magnetic field of the electrons at the nucleus is *) T h e s p e c t r a l lines are shifted in the s e c o n d order b y a m o u n t ~
(q2/B).
HYPERFINE STRUCTURE IN PARAMAGNETIC RESONANCE
183
finite. There are four such ions, namely vanadium, manganese, cobalt and copper, which have been studied as divalent ions in either tutton salts, such as Mn(NH4)2(SO,)2, 6H20 or fluosilicates MnSiF6, 6H20. The former have two ions in unit cell, each with approximately tetragonal symmetry; the latter have only one, with tfigonal symmetry. To obtain narrow lines, highly "diluted" salts in the form of mixed crystals with the isomorphous zinc salt, have been grown. At concentrations of the order of one paramagnetic ion to about a thousand zinc ions, a limiting half-width at half intensity of about 7 gauss is attained. This residual width is attributed to the twelve protons in the waters of crystallization which lie immediately around the paramagnetic ion. The hyperfine structure of these four ions contained significant differences in behaviour, and it is convenient to consider them separately. Manganese. 3d s, 6S, I s/2. The manganese ion serves as a test point of theory since the 3d shell is just half-filled, so that the free ion is in a s S state and no crystalline field is required to "quench" the orbital motion. As a result, the value of g (2.000 4- 0.001 in the fluosilicate) is very close to the free spin value (2.0023,) and exceedingly isotropic. A small splitting 4) of the 6S state is found, corresponding to values of D respectively of + 0.027 cm -1 (manganese ammonium sulphate) and --0.013 to --0.018 cm - I (fluosilicate, depending on temperature). These splittings are attributed to dipole-dipole interaction between the spin moments of the five 3d electrons; this interaction is zero if the electron cloud has spherical symmetry, as in the free ion, but is not zero if a small axial distortion is present due to the crystalline field 5). As a consequence of this spherical symmetry, the magnetic field and electric field gradient at the nucleus would be zero for the free ion, and no his would result. The small distortion in the solid could produce only a tiny his, with A = - - 2B, and both of the order of 10-4 cm -L. In fact, an almost isotropic his is observed, of magnitude one hundred times greater. It is practically identical in magnitude in the two salts, showing that it is not due to the crystalline field, since the electronic splitting shows this differs in magnitude (and sign) in the two salts. A b r a g a m e ) has pointed out that an his of this magnitude must be due to an unpaired s-electron; he suggests that =
184
B. B L E A N E Y
the .3s23p63d~, 6S state h a s admixed it some of a 3s3d63dS4s, 6S state, in which a 3s electron is promoted to the 4s level. Only a very small admixture of such a state is required, since a 3s electron is very plotent in producing hyperfine structure. The admixture is due to the electrostatic repulsion of the electrons (configurational interaction), Which is ~, property of the free ion.,This explains why the effect is the same in two very different salts.
Vanadium, 3d a, I =
7/2.
H e r e the free ion is in a 4/~ state, but a crystalline field of cubic symmetry quenches the orbit very effectively. The lowest orbital level is a singlet, some 10,000 cm ~l below any other orbital level. The value of g is 1.96, and is isotropic. This lowest level is fourfold in the Spin, but is split into two doublets (in zero magnetic field) b y second order spin-orbit interaction and the spin-spin interaction described far manganese. In vanadium ammonium sulphate this splitting corresponds to D----0.16 cm -I, but there is also a very considerable rhombic contribution. The wave function of the lowest orbit is xyz, which has six equal lobes along the rectangular axes, This leads to zero magnetic field at the nucleus from the electron spins, as m a y easily be seen classically b y computing the field at the origin due to six dipoles placed at equal distanoes along the axes. There remains only the orbital contribution, which s h o u l d b e small as the departure of g from the free spin values is small. Experimentally a wide and iso~ropic his is observed, with A = B = 0.01 cm -I, which is practically the same as in manganese. Though this is only a coincidence, there is no doubt that the main contribution to this his comes again from admixture with a state in wich a 3s electron is promoted. The spin of 7/2 (K o pf e r m a n n and R a s m u s s e n S ) ) , which is given with a query in some tables, is confirmed. Some preliminary work has been carried out to determine the spin of chromium 53, which is present in 9 o abundance in the natural state, using a diluted chrome alum. Here the chromium trivalent ion isiso-electronic with divalent vanadium. No h/s could be observed and it seems probable that the overall width of any h/s must be less than 60 gauss, as compared with 680 in vanadium. This would suggest that the nuclear moment of Cr~ is less than 0.5 nuclear magnetons, but further work is in hand to confirm this result,
HYPERFINE
STRUCTURE
IN PARAMAGNETIC
RESONANCE
185
Cobalt, 3d 7. Co59~I, = 7/2. As in the case of V ++ , the state of the tree ion is 4F, but as we are dealing with three holes in a full shell instead of three electrons, the splitting of the orbital levels of cobalt b y the major (Cubic) portion of the crystalline field is inverted. The cubic field leaves an orbital triplet at t h e bottom, which is split .up b y spin-orbit interaction and tetragdla~(~(:or trigonal) crystalline field 9) to a' rather small degree. The lowest is a K r a m e r's doublet, a n d t h e next level lies only a few hundred cm -1 higher. This accounts for the great anisotropy shown b y cobalt salts, whose susceptibility also departs widely from C u r i e ' s law. The results obtained with three salts are summarized b y the values lo) given in the following table. ,, TABLE I g-values of Co
.
A , cm - t
B em -~
3.44
0.018
0.005
6.55
2.4-3.3
0.029.
0.007
6.45
3.06
0.025
0.002
Salt
g#
C o b a l t fluosilicate . . Cobalt potassium sulphate Cobalt ammonium sulphate .....
5.82
g±
The variation in g± shown for the potassium sulphateis due to the presence of rhombic asymmetry; the other salts have very accurately axial symmetry. It will be seen that both the g-values and the h/s show considerable anisotropy. P r y c e and A b r a g a m n), in an extensive study of cobalt, have shown how the theory which accounts for the g-values gives also a reasonable approximation to the hyperfine structure. The latter is due almost entirely to interaction with the orbital moment, the spin contribution being only about 10°/o. Some interaction with a state in which a 3s electron is promoted is required to explain the high anisotropy, but the effect of such a state is relatively small. Copper, 3d 9, 2 D . C u 63, C u 6s, I ~--- 3/2. In a field of tetragonal symmetry, as in the tutton salts, the orbital m o m e n t u m of Cu + + is largely quenched, as is shown b y the values of g. ParaUel to the axis, g is 2.4---2.5; perpendicular, g is 2.05--2.1. This suggests that the his will be due primarily to interaction with
186
B. BLEANEY
the electron s p i n , and at first sight the measured anisotropy (A ---- 0.012 cm - l , B ---- 0.001 to 0.003 cm -1, varying with the salt) supports this. The lowest orbital wavefunction is ( ~ - y2), which has four equal lobes along the two axes normal to the tetragonal axis. If four dipoles are placed at equal distances along these axes, it is easily seen that the field which they produce at the origin is twice as great when the dipoles are aligned parallel to the z-axis, as it is when they are aligned in the xy-plane. As there is also a change of sign, one would expect A ~ - - 2 B ; this appears to fit the results reasonably, since the signs are indeterminate (see § 5). A b r a g a m and P r y c e 12,e) have shown however that the contribution from the unquenched orbit upsets this pseudo-agreement completely, giving A = - - 0 . 0 0 6 cm - I , B = 0.013 cm -I. Theory can only be reconciled with experiment b y admixing excited configurations; 3d84s 2D gives a contribution whose sign is ~uch as to worsen the situation, but 3s 3p63d94s, 2/9 provides the right correction. A b r agame) shows that one can obtain then A - ~ - - 0 . 0 1 5 cm -1, B ~ + 0.004 cm -1 in reasonable agreement with experiment. Thus again a significant feature of the theory is that a state in which an inner 3s electron is promoted to a 4s must be considered. Although copper has two abundant isotopes, their magnetic moments differ only b y 7%, and their his shows only partial resolution when it is widest, near the axis. I n g r a m 18) discovered, however, that the spectrum is split into more than four lines, when the magnetic field is nearly perpendicular to the axis. P r y c e has shown that the general behaviour can be fitted b y assuming the existence of a quadrupole effect with Q ~ 0.001 c m - ' . It is hoped b y further work on this spectrum to obtain more exact information of the size of Q for the two isotopes, 63 and 65. As this is the only case in which a quadrupole effect has been detected, it should be pointed out that this does not imply that the electric quadrupole moments of vanadium, manganese and cobalt are necessarily much smaller than those of the copper isotopes. It is obvious that the electron cloud distribution for V and Mn is too symmetrical to give an appreciable gradient of the electric field at the nucleus, and the admixture of an excited state with an unpaired s-electron does. nothing to change this situation. In cobalt also, theory indicates that any quadrupole interaction would be small.
HYPERFINE
STRUCTURE
IN
PARAMAGNETIC
RESONANCE
187
From the experiments only the following upper limits can be set for Q : vanadium, manganese (divalent salts), 0.00 1 cm-’ cobalt (divalent salts), 0.0005 cm-‘. 4. Hyfierfine structure in the rare earth group. Little paramagnetic resonance work has been carried out so far on salts of the rare earth group, but a start has recently been made at Oxford. One of the most suitable salts seems to be the ethyl sulphates, which have trigonal symmetry, and, from the magnetic point of view, only one ion in unit cell. These are already, magnetically, fairly dilute, and can be diluted further by the isomorphous salt of lanthanum. The results obtained thus with nedodymium and gadolinium are given below. The rare earth group differs from the iron groups in that L and S still combine to form a resultant J, and the crystalline field is strong enough only to split up the 2J + 1 levels corresponding to the different orientations of J. These splittings are thought to be of the order of IO-1 00 cm-‘. Although,when J is half integral, the crystalline field must leave at least double degeneracy in the levels, it does not necessarily follow that resonance absorption can be obtained in the region V = 1 cm-‘. The degenerate levels may correspond, for example to J, = + 3/2 and - 3/2,between which no transitions are allowed. However, it seems probable, that by choice of a salt with suitable crystalline field, absorption lines may be obtained in all these cases, This is important, for the trivalent ions with K r am e r s degeneracy are cerium, neodymium, samarium, gadolinium dysprosium, erbium and ytterbium. Though these elements have even atomic number, all except cerium have one or two odd isotopes of natural abundance 1O-2Oo/o,and the spins of seven of these are not previously known. Work has therefore been concentrated on these, starting with neodymium and gadolinium l*). Neodymizrm 4f3. Odd isotopes : 143, 12.2% abundance. 145, 8.3% abundance. Even isotopes : 79 y0 abundance. An extensive hyperfine structure in concentrated neodymium ethyl sulphate was reported by B 1 e a n e y and I n g r a m lo). At a dilution of Nd : La w 1 : 200, the half width of the lines is reduced to about 7 gauss. The hyperfine structure is then completely resolved, and two series each containing eight equally spaced lines
188
B. B L E A N E Y
can be assigned to the odd isotopes 143, 145. F r o m t h e spacing the ratio of the m a g n e t i c m o m e n t s (143/145) is f o u n d to be 1.61 + 0.01. T h e d a t a m a y be s u m m e d up as follows. g// =
3.58,
gL
: 2.09
Isotope 143 : A = 0.040 c m -1, B = 0.020 cm - t , I = 7/2. Isotope 145: A = 0.025 cm -1, B = 0.012 cm - j , I = 7/2. T h u s the anisotropy is 2 : 1 in the hyperfine structure, a n d a b o u t 1.7 : I in the g-values. No effects due to a q u a d r u p o l e interaction could be observed, a n d an upper limit for {) of 0.002 cm -I can be assigned for either isotope. Gadolinium, 4/7. Odd isotopes: 155, 15% aburndance. 15•, 15% abundance. E v e n isotopes: 70% abundance. The trivalent gadolinium ion is in an 8S state, and a fine s t r u c t u r e is observed due to a small splitting of the levels. Like manganese, this splitting is due primarily to an axial distortion ; the value of t h e p a r a m e t e r D is a p p r o x i m a t e l y 0.023 cm -1, varying by about 105/o between room t e m p e r a t u r e and 20°K. Although t h e signal from each line of the fine s t r u c t u r e was 50 to 100 times noise, no weak lines were visible belonging to h y p e r f i n e structure. As a f u r t h e r check the s p e c t r u m was observed at an angle where 5 of t h e fine s t r u c t u r e lines coincide; since a n y hfs belonging to these transitions would also coincide. This should give a gain of five in intensity; its absence showed conclusively t h a t there was no resolved h/s. The overall w i d t h of a n y his m u s t be less t h a n 40 gauss, c o m p a r e d with a maxim u m w i d t h of 1600 gauss in n e o d y m i u m . This absence of hfs shows t h a t t h e g r o u n d state of the gadolinium ion is exceedingly free from a d m i x t u r e of a n y excited states in which, for instance, a 4s or 5s electron has been p r o m o t e d to the 6s level. T h u s the situation is v e r y different from the iron group, where the interaction w i t h such excited states is sufficient to produce an his in v a n a d i u m a n d manganese (where there should be none) of the same order or greater t h a n in cobalt and copper. As a result, no information concerning the nuclei of the odd gadolinium isotopes could be obtained.
5. Conclusion. At t h e end of this brief review of the work on his in the solid state carried out at the Clarendon L a b o r a t o r y in the past 20 months, it seems appropriate to discuss the possibilities of ob-
H Y P E R F I N E STRUCTURE IN P A R A M A G N E T I C RESONANCE
189
taining nuclear information b y this method. It is, of course, limited to atoms which form paramagnetic salts, but it is just these which are rather inaccessible to other methods. It is obvious that a simple and direct determination of nuclear spin is possible, together with a fairly accurate measure of the ratio of magnetic moments of two isotopes of the same atom. The h/s splitting can be measured with an accuracy (at present) of about 1% ; this could be improved b y the use of a proton resonance calibration for the magnetic field. There seems little gain in such improvement, however, because of the difficulties of the theoretical interpretation. It is well known that high accuracy is not obtainable in the calculation of absolute values of nuclear moments from optical h/s; here one has the additional complication of the crystalline .field. In the iron group, it appears' :that configurational" interaction plays a very important role, and the theory which is adequate to explain the g-values tells us practically nothing about this effect. In the rare earth group, the absence of h/s in gadolinium suggests that this effect may be much less troublesome, b u t here the theory of the g-values is rudimentary. K. W. H. S t e v e n s 16) has tackled the neodymium ion, and we may hope that when a suitablecombination of crystalline field and spin-orbit interactions has been found to fit the g-values, a reliable estimate of the nuclear moments m a y also be obtained. A regrettable feature of the method is that it gives no information about the actual signs of the constants A, B, Q. Where several electronic transitions are observed, as in manganese, the relative signs A, B and D can be found from second order effects of the spectrum in not quite strong field; B 1 e a n e y and I n g r a m 4) have combined this with a measurement of the anisotropy of the susceptibility which gives the sign of D, to obtain all the signs. But when S = ½, not even the relative signs of A, B can be found in the simple case. Where a quadrupole effect is observed, as in copper, the extra transitions can m theory be used to find the relative signs, when allowance is made for the direct effect of the external field on the nucleus. The second order effects in copper potassium sulphate suggests that B, q have the same sign, b u t even here a previous knowledge of the sign of nuclear moment is necessary. More accurate information about the size of q, and its sign relative to A can be obtained from observations of the spectrum in .zero or very small magnetic field, and work in the region of 50--100 cm wavelength is planned. Received 23-9-50.
] 90
HYPERFINE STRUCTURE IN PARAMAGNETIC RESONANCE REFERENCES
1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) IS)
K i t t e l , C.H., Phys. Rev. 76 (1949) 743. Gorter, C . J . , Physica, 14 (1948) 504. Penrose, R . P . , Nature Lond., 163 (1949) 992. Bleaney, B. and I u g r a m , D. J. E. Proc. roy. Soc. A205(1951) 336. Abraham, A. and P r y e e , M . H . L . , P r o c . roy. Soc. A'205(1951) 135. Abragam, A., Phys. Rev. 79 (1950) 534. I n g r a i n , D . J . E . a~d S e o v i l , H . E . D . , (to be published). Kopfermann, H. and R a s m u s s e n , E., Z. Physik, 98 (1936)624. P r y c e , M . H . L . , NatureLond, 164(1949) 116. Bleaney, B..and I n g r a m , D . J . E . , NatureLond. 164(1949) l l6 and future publication. Abragam, A. and P r y c e , M. H. L. (to be published). Abragam, A. and P r y c e , M . H . L . , Nature Loud. 163 (1949) 992. I n g r a i n , D. J. E., Prox. Phys. Soe. A6'2 ¢q949) 664. Bleaney, B. and S e o v i l , H . E . D . , Proe. phys. Soc. A 63 (1950)1369. Elliott, R. J. and S t e v e n s , K . W . H . , P r o e . phys. Soc. A64(1951) 205.